Probability of winning a scratch lottery ticket given that 95% have been sold

[HokieBalla34]

A buddy from work and I have been debating an article we read the other day stating that it is a bad investment to play scratch lottery ticket games that have been on the market for an extended period of time (ex: hypothetical 2004 Christmas Game Scratcher). The article stated that this is due to the fact that the probability that the winning jackpot ticket has already been sold. My buddy concurs with this, but it doesn't sit well with me. I would argue that this is no worse of an investment (buying in the latter half) than if you had bought the very first ticket. Could someone please help?

We have made a few logical assumptions:
1. 95% of the tickets have already been sold
2. It is not known whether or not the winning jackpot ticket has already been sold
3. There is only one winning jackpot ticket

He states that since you are buying in the last 5% of tickets that you are making a poor investment. I counter that this could be said no matter what 5% you are in (even first 5% because the ticket is more likely to be sold in the latter 95% by that argument). I also understand that the odds of purchasing the winner are 1/n, where n represents the number of tickets created. However, I am not entirely positive (mathematically) why it would still be ok to buy in the latter 5%. Thanks in advance for any information!

 

[chroot]

Here's the deal: there are two distinct cases to consider.

1) You don't know whether or not the winning ticket has already been sold. Since the probability of winning is equal for every ticket, you have the same probability of winning whether you buy the first or the very last ticket.

2) You do know whether or not the winning ticket has been sold. If so, the probability of any later ticket winning is exactly zero, and it would be bone-headed to play the game.

In reality, you can probably look up lottery winners in some kind of a public database (perhaps even online in your locality). If anyone has won the jackpot, don't bother playing the game. On the other hand, if no one has won the jackpot, but only 5% of the tickets remain to be sold and one of them is a winner, your probability of winning per ticket is much greater than it was when the game was first introduced.

- Warren

 

[DaveC426913]

Two situations:

1] If there is 1 [one] pick, then you are correct. The likelihood that the winning ticket is the very last ticket bought is exactly the same as if it were any other ticket.

2] I think the article is referring to the fact that, in some lotteries, the big prizes are picked, and yet they continue to sell tickets for other prizes. This is something that hit the news a while back. In this case, the odds of winning the big prizes are zero.

 

[HokieBalla34]

Yes, I understand how to treat it as an individual problem where the probability is 1/n. That is a trivially simple case. And as I said, you are going on the fact that you do not know if the winning ticket has been sold or not--therefore it does not matter whether or not the lottery game still plays out after the jackpot is hit.

It seems like a trivially easy problem until you think about it in terms of good investment vs. poor investment:

I feel that concrete knowledge that 95% of the tickets have been sold (without knowing if one of those were a winner) should not bar you from buying one out of the remaining 5% of tickets. He feels that it should.
His argument is that for each ticket sold, buying a ticket becomes a poorer and poorer investment. His argument is that there is 95% chance that the winning ticket has been sold to one of the first 95% of buyers, so there is greater chance that the ticket has already been sold and you are wasting your money. I do not agree with this and feel that your investment is the same no matter how many tickets are remaining.

I am interested in hearing solid mathematical arguments from both sides, so fire away!

 

[jrover112]

Wrong

I believe you all are answering the question using a false assumption that the winner is picked upon all tickets being sold. This is wrong due to the fact that it is not a raffle type lottery. The initial question stems from an article concerning scratch off tickets.

Under the circumstances above, I stand by my initial argument (I am the originators buddy from work) that there is a greater chance that the winning ticket has already been sold under the assumption that the majority of tickets have been bought (say 95/100 total tickets for simplicity).

 

[DaveC426913]

Can we lay this out step-by-step, so there are no assumptions?

This may or may not be accurate:
100 tickets are printed up.
1 (and only one) ticket is a winner.
Ticket purchase closes before the winning number is drawn.
The first ticket sold has a 1/100 chance of being the winner.
The 94th ticket sold has a 1/100 chance of being the winner.

 

[DaveC426913]

jrover, perhaps the piece you are missing is this:

The last ticket buyer is 99% positive that the winning ticket HAS been purchased. You're right - it is very low odds he will get the winner (only 1 chance in 100 that the last ticket is the winner).

On the other hand, the FIRST ticket buyer is 100% guaranteed that the winning ticket has NOT been purchased yet. Excellent odds, right?

Until he realizes that he has 99 chances in 100 of picking the WRONG ticket. In fact, the chance that the first ticket is the winning ticket - is only 1 in 100.

What balances out is that, whether 1 ticket or 95 tickets or all 100 tickets are purchased, each individual ticket - be it #1, #49, #95 or #100 - has exactly 1 in 100 chances of being the winner.

 

[HokieBalla34]

Wrong

I believe you all are answering the question using a false assumption that the winner is picked upon all tickets being sold. This is wrong due to the fact that it is not a raffle type lottery.

Actually, the stated assumption that you do not know whether or not the winning ticket has already been sold essentially makes it a raffle. In a raffle you have no clue who the winner will be until all tickets are sold and one is drawn. The probability of one matching ticket being drawn in a raffle is the exact same probability of hitting one jackpot in 'n' printed tickets (assuming equal number of tickets in each). If everyone bought a ticket and did not peel it until all tickets have been sold (silly, but to put a better perspective on the ticket chances), you would have exactly the same situation as a raffle. The person who bought in 96th place would be just as likely to win as the person in 1st place, even though 95 tickets had been sold in front of him. The number of tickets that have already been sold does not affect the odds that the ticket you hold in your hand is the winning ticket.

 

[chroot]

Bottom line: if you don't know whether or not a winner has been found already, then every ticket (including the very last one) is an equally good "investment." That situation, however, is artificial.

- Warren

 

[jrover112]

Wrong again

The statement in the article was as follows:
Do not purchase a scratch off ticket if the game has been going on for a substantial amount of time because there is a good chance the winning ticket has already been sold.

Look at it this way:
1. 95% of tickets have been sold
2. The probability of each ticket being sold is 1/n
3. As you increase the number of tickets sold, you increase the probability that the winning ticket has been sold.
4. In other words if I have 95 chances out of 100 to buy the winning ticket, there is a better chance that I bought the winning ticket than if I had 5 chances out of 100.

 

[0rthodontist]

What if you have, as you word it, "99 chances out of 100" to buy the winning ticket? Are you saying that the first person to buy a ticket nearly always wins? This is not true.

When you buy a ticket, you get one chance out of 100, no more, no less, unless you have extra information like "no one has won yet" or "someone has already won."

Yes, when you increase the number of tickets sold you increase the probability that the winning ticket has been sold--so if you buy the 96th ticket there is a 95/100 chance that the winning ticket has already been sold. There is thus a 5/100 chance that the winning ticket has not yet been sold, and IF it hasn't been sold, then there is a 1/5 chance that you bought the winning ticket. Your probability of winning the ticket is thus the probability that the winning ticket had not yet been sold, times the probability that, given that the winning ticket had not yet been sold, you are the one who got it. This is 5/100 * 1/5 = 1/100. If you buy the 6th ticket there is only a 5/100 chance that the winning sicket has already been sold, so that there is a 95/100 chance that the winning ticket had not yet been sold. And there is a 1/95 chance that, IF the winning ticket had not yet been sold, you got it. Your probability of getting the winning ticket is thus the probability that the winning ticket had not yet been sold, times the probability that, given that the winning ticket had not been sold, you are the one who got it. This is 95/100 * 1/95 = 1/100, the same chance as for the 96th ticket.

 

[HokieBalla34]

Point taken; however, I don't see any way it can be proven that P[winner=96] (for example) is dependent on the 95% chance that the winning ticket no longer remains. Same said for P[winner=3] given that 96% the winner remains. I would have to see something along the lines of P[winner=96] = 1/n - P[ticket has been sold] = 1/100 - .95 = -.94 (an obviously impossible result) in order to believe that the 95% probability the ticket has already been sold affects my 1% odds.

 

[DaveC426913]

1. 95% of tickets have been sold
2. The probability of each ticket being sold is 1/n
3. As you increase the number of tickets sold, you increase the probability that the winning ticket has been sold.
4. In other words if I have 95 chances out of 100 to buy the winning ticket, there is a better chance that I bought the winning ticket than if I had 5 chances out of 100.

If you have 95 chances out of 100 to buy the winning ticket, you have a 94/95 chance of picking the WRONG one.

Rationalize as you might, in a standard lottery, the odds on the 1/100th ticket being the winner are exactly 1 in 100.


I ask again for clarification on the type of lottery.
Is there only ONE prize?
Is there only ONE winning card?*

*(Note that, with a scratch-off type lottery, it is possible for there to be only one prize, but *several* winning cards. This changes the scenario dramatically because it introduces a "first come first served" rule.)

 

[HokieBalla34]

In our scenario,there is only one grand prize and only one winning card. We are not considering the case of multiple prizes, however, I do not think this situation would be any different (other than increasing your odds across the board of purchasing some sort of winner). Your chances of hitting ANY ONE prize does not change depending on number of tickets sold before your purchase--I am not sure what you mean by a "first come first serve" rule.

 

[DaveC426913]

I am not sure what you mean by a "first come first serve" rule.

I've been trying to eke out the conditions of the lottery.

Some lotteries are like bingo - the first winner gets the prize, even if subsequent people get he same numbers. Clearly, this is not one of those types.

However:

Some lotteries have the main prize go, but they still continue to sell tickets for the smaller prizes. I was hypothesizing that the lottery written about in the the article you refer to was this kind of lottery.

You see, this was an newsworthy item a few years back. In certain lotteries, peole were unaware that the tickets they were buying could only win them a lesser prize; the main prize had already been doled out.

Are you guys are certain that this is NOT the kind of lottery referred to in the articles?

 

[DaveC426913]

In our scenario,there is only one grand prize and only one winning card.

Then you are correct and jrover is wrong.

 

[shmoe]

It would help if you could provide a reference to the original article but as I understand the assumptions that have been presented here, the conclusion of the article is false- all tickets are equally poor investments.

 

[HokieBalla34]

It would help if you could provide a reference to the original article

http://moneycentral.msn.com/content/Retirementandwills/Retireearly/P89288.asp

Here is an excerpt from the article, under HOW TO PLAY SMARTER
Beware the stale game.
People often don’t realize that scratch games aren’t finished when someone wins the biggest prize; the tickets are left out until they’re all sold. That means you might be buying a ticket to a game in which there’s no chance for a juicy payday, says Chris Gudgeon, co-author of “Luck of the Draw: True-Life Tales of Lottery Winners and Losers.” Gudgeon’s advice: Avoid scratch games that have been lingering near the Slurpee machine for ages. “If you’re buying the scratch-and-wins, particularly the seasonal ones, don’t buy a Christmas one at the following Halloween,” he says. “There’s a very good chance that all of the prizes are gone.”

Agreed, there is a VERY GOOD CHANCE that the prize is gone. But as this thread provides, since you don't know whether or not it remains, you should not be dissuaded from purchasing one of the remaining ones (Whether any ticket purchase is a good investment is not at all relevant to this discussion).


Thanks all for your input.

 

[DaveC426913]

People often don’t realize that scratch games aren’t finished when someone wins the biggest prize; the tickets are left out until they’re all sold...

THANK you. Durrr.

I've been trying to clarify that point all along: tickets are sold AFTER the big prize is won. Posts https://www.physicsforums.com/showpost.php?p=924617&postcount=3" explicitly ask about this particular condition, or state it as an assumption.

That invalidates virtually ALL previous discussion about the odds (except mine that is hee.)

 

[daveb]

Look at it this way:
1. 95% of tickets have been sold
2. The probability of each ticket being sold is 1/n
3. As you increase the number of tickets sold, you increase the probability that the winning ticket has been sold.
4. In other words if I have 95 chances out of 100 to buy the winning ticket, there is a better chance that I bought the winning ticket than if I had 5 chances out of 100.

I believe your argument is that if there are n tickets that can be sold, the probability of any single ticket winning is 1/n. However, if m<n tickets have been sold, the probability the winning ticket has already been sold (assuming you don't know if this is the case) is m/n, or P(winner <=m). Is that what you mean?

 

[0rthodontist]

THANK you. Durrr.

I've been trying to clarify that point all along: tickets are sold AFTER the big prize is won. Posts https://www.physicsforums.com/showpost.php?p=924617&postcount=3" explicitly ask about this particular condition, or state it as an assumption.

That invalidates virtually ALL previous discussion about the odds (except mine that is hee.)

No, it still doesn't make any difference. Unless you actually KNOW of particular prizes in the game that have been sold already, it doesn't matter when you get the ticket, even if there are a thousand small prizes.

If you did somehow know what prizes were given out, and you noticed that there were a disproportionately small or large number of them for the number of people that have played the game, then that would be one situation where you actually could invest wisely by either not buying a ticket (if you know that disproportionately many prizes have been given out) or buying a ticket (if you know that disproportionately few prizes have been given out). But that happens extremely infrequently, ESPECIALLY when there are many small prizes.

If there is one big prize that someone already got, and even it's shown on the news that they got it, but so long as you didn't see that news report and you buy a ticket--then from the knowledge you have your odds are STILL the same as they were as if the news had not been shown.

 

[HokieBalla34]

THANK you. Durrr.

I've been trying to clarify that point all along: tickets are sold AFTER the big prize is won. Posts https://www.physicsforums.com/showpost.php?p=924617&postcount=3" explicitly ask about this particular condition, or state it as an assumption.

That invalidates virtually ALL previous discussion about the odds (except mine that is hee.)

DURRRR and post 1 states that a few logical assumptions have been made, namely that you don't know if the winning jackpot ticket has been sold or not. Therefore, it is a moot point whether or not the game is still played out after the ticket has been won.

 

[DaveC426913]

DURRRR and post 1 states that a few logical assumptions have been made, namely that you don't know if the winning jackpot ticket has been sold or not.

Of course you don't know if the winning ticket's been sold! The only way you could know that is if you're clairvoyent or cheating. Which is why is was a pointless assumption.

Or did you mean the winning ticket has already been announced?

 

[HokieBalla34]

Hence the word LOGICAL. This assumption was made merely to dissuade people who would inevitably make the argument that your chances would vary depending if you knew the winner had been sold (or not been sold). Since this information would not be relevant to our debate (which is whether the fact that 95% of the tickets had been sold affects your present odds, absent any other POINTLESS information), the assumption was made in order to determine the desired answer and eliminate extraneous debates/information.

And on a side note, you couldn't be more incorrect with your statement: "Of course you don't know if the winning ticket's been sold!" It is actually VERY simple for one to determine if the ticket had been sold--using the right resources, of course (Hint: It doesn't involve enlisting Mrs. Cleo or a cheat!). I may be letting the cat out of the bag here but it's called THE INTERNET An example of what I am talking about may be easiest for you to understandhttp://www.valottery.com/news/press_details.asp" . I would imagine that Virginia isn't the only state that publishes their results but did not care to check. That may be a good exercise for the Santa Barbara School of the Clairvoyant or Cheaters Anonymous.

And no, I did not mean "the winning ticket has already been announced". Obviously you wouldn't assume that "the winning ticket has already been announced" because that would imply that the winning ticket no longer remains and P[x=winner] = 0. Talk about pointless assumptions.

But thanks everyone for your input. The situation has been clarified and the debate is closed.

 

[DaveC426913]

And on a side note, you couldn't be more incorrect with your statement: "Of course you don't know if the winning ticket's been sold!" It is actually VERY simple for one to determine if the ticket had been sold--using the right resources, of course (Hint: It doesn't involve enlisting Mrs. Cleo or a cheat!). I may be letting the cat out of the bag here but it's called THE INTERNET An example of what I am talking about may be easiest for you to understandhttp://www.valottery.com/news/press_details.asp" . I would imagine that Virginia isn't the only state that publishes their results but did not care to check. That may be a good exercise for the Santa Barbara School of the Clairvoyant or Cheaters Anonymous.

And no, I did not mean "the winning ticket has already been announced". Obviously you wouldn't assume that "the winning ticket has already been announced" because that would imply that the winning ticket no longer remains and P[x=winner] = 0. Talk about pointless assumptions.

That was the clartification I've been asking for. A bit more clear than the initial one sentence.

 

[RyanPMcBride]

I just saw this post... even though it is quite old I wanted to input some statistics into the solution.

1. If you buy the 1st ticket sold out of 100 your chances are 1/100
2. If you buy the 44th ticket sold out of 100 your chances are now 44/56 assuming the winning ticket has not yet been sold. It is 56 because 44 tickets have already been sold that have not won (100-44).
3. The problem with this theory is that you will never know when the winning ticket is sold. But if you somehow did find out there were no winners yet, then obviously your chances are greater.

Just thought I would throw that in there.

Peace!

 

[DaveC426913]

2. If you buy the 44th ticket sold out of 100 your chances are now 44/56 assuming the winning ticket has not yet been sold.

This is wrong (as I'm sure you will quickly realize. Having a 44/56 chance of winning is ridiculously good odds.)

If you buy the 44th ticket out of 100 (and no ticket yet was the winner) your chances of being the winner are 1/57. (You have an equal chance of winning as any of the 56 subsequent players.)